中文摘要：

先把求解微分方程的周期解问题转化为无约束最优化问题,再利用无约束最优化问题的最优性条件及Levenberg-Marquardt方法求解了满足限制共振条件下的一类Duffing方程的周期解.数值计算结果表明了方法的有效性.

英文摘要：

We converted the periodic solution problem of solving differential equations to the unconstrained optimization problem.With the help of the optimality conditions of unconstrained optimization problem,we applied the Levenberg-Marquardt method to finding the periodic solutions of a class of Duffing equations under restrict resonant conditions.